On the topology of the magnetic lines of solutions of the MHD equations

Abstract

We construct examples of smooth periodic solutions to the Magnetohydrodynamic equations in dimension 2 with positive resistivity for which the topology of the magnetic lines changes under the flow. By Alfv\'en's theorem this is known to be impossible in the ideal case (resistivity = 0). In the resistive case the reconnection of the magnetic lines is known to occur and has deep physical implications, being responsible for many dynamic phenomena in astrophysics. The construction is a simplified proof of [3] and in addition we consider the case of the forced system.